Game list colouring of graphs (Q873968)

Summary: We consider the two-player game defined as follows. Let \((G,L)\) be a graph \(G\) with a list assignment \(L\) on its vertices. The two players, Alice and Bob, play alternately on \(G\), Alice having the first move. Alice's goal is to provide an \(L\)-colouring of \(G\) and Bob's goal is to prevent her from doing so. A move consists in choosing an uncoloured vertex \(v\) and assigning it a colour from the set \(L(v)\). Adjacent vertices of the same colour must not occur. This game is called game list colouring. The game choice number of \(G\), denoted by \(\text{ch}_g(G)\), is defined as the least \(k\) such that Alice has a winning strategy for any \(k\)-list assignment of \(G\). We characterize the class of graphs with \(\text{ch}_g(G)\leq 2\) and determine the game choice number for some classes of graphs.